Molecules in solution are generally characterized by their weight averaged molar mass Mw, their mean square radius       ⟨          r      g      2        ⟩    ,
and the second virial coefficient A2. The latter is a measure of the interaction between the molecules and the solvent. For unfractionated solutions, these properties may be determined by measuring the manner by which they scatter light using the method described by Bruno Zimm in his seminal 1948 paper which appeared in the Journal of Chemical Physics, volume 16, pages 1093 through 1099. The light scattered from a small volume of the solution is measured over a range of angles and concentrations. The properties derived from the light scattering measurements are related through the formula developed by Zimm:
R(xcex8)=K*Mw,cP(xcex8)[1xe2x88x922A2Mw,cP(xcex8)]+O(c3),xe2x80x83xe2x80x83(1)
where R(xcex8) is the measured excess Rayleigh ratio in the direction xcex8 per unit solid angle defined as R(xcex8)=[Is(xcex8)xe2x88x92Isolv(xcex8)]r2/I0V, Is(xcex8) is the intensity of light scattered by the solution a function of angle, Isolv(xcex8) is the intensity of light scattered from the solvent as a function of angle, I0 is the incident intensity, r is the distance from the scattering volume to the detector, V is the illuminated volume seen by the detectors, P(xcex8) is the form factor of the scattering molecules defined as P(xcex8)=limcxe2x86x920R(xcex8)/R(0), K*=4xcfx802(dn/dc)2n02/(Naxcex04), Na is Avogadro""s number, dn/dc is the refractive index increment, n0 is the solvent refractive index, and xcex0 is the wavelength of the incident light in vacuum. The form factor is related to the mean square radius by                               P          ⁢                      xe2x80x83                    ⁢                      (            θ            )                          =                  1          -                                                    16                ⁢                                  xe2x80x83                                ⁢                                  π                  2                                ⁢                                  xe2x80x83                                ⁢                                  n                  0                  2                                                            3                ⁢                                  xe2x80x83                                ⁢                                  λ                  0                  2                                                      ⁢                          xe2x80x83                        ⁢                          ⟨                              r                g                2                            ⟩                        ⁢                          xe2x80x83                        ⁢                          sin              2                        ⁢                          xe2x80x83                        ⁢                          (                              θ                /                2                            )                                +                      O            ⁢                          xe2x80x83                        ⁢                          (                                                sin                  4                                ⁢                                  xe2x80x83                                ⁢                                                      (                    θ                    )                                    /                  2                                            )                                                          (        2        )            
The collection of light scattering data over a range of scattering angles is referred to more commonly as multiangle light scattering, MALS. The data are then fit to Eq. (1) to extract Mw,       ⟨          r      z      2        ⟩    ,
and A2. For this purpose, the reciprocal of Eq. (1) is more commonly used which may be written as:
K*c/R(xcex8)=1/[MwP(xcex8)]+2A2c+O(c2).xe2x80x83xe2x80x83(3)
There are several methods by which the data may be fit to Eq. (3). The most popular, historically, is to mimic in software the graphical method presented by Zimm. This is often referred to as the Zimm Plot method. Alternatively one may use the global nonlinear least squares fit described below.
A powerful method of characterizing a molecular solution is to fractionate the sample first by chromatographic means, such as size exclusion chromatography, SEC, and then measure the scattered light and concentration as a function of elution volume xcexd. If the fractionation is sufficiently resolved, each volume sample can be considered to be essentially monodisperse. If A2 is known from prior experiment, or the concentrations are low enough that the effect of A2 on the scattered light is negligible, one may fit the data to Eq. (1) or (3) to extract the distributions M(xcexd) and rg2 (xcexd). This is routinely performed by commercial software such as the ASTPA(copyright) program developed by Wyatt Technology Corporation of Santa Barbara, Calif.
One may define a figure of merit, FOM, which characterizes when the A2 term may be neglected. Equation (1) shows that A2 is the prefactor of the c2 term. By comparing the magnitude of the bracketed terms in Eq. (1), the FOM may be defined as
FOM=2A2Mwc.xe2x80x83xe2x80x83(4)
When the FOM less than  less than 1, as was assumed in the derivation of the Zimm equation, the second virial coefficient has only a small effect on the light scattering signals. When one wishes to measure the second virial coefficient by light scattering, the FOM must be large enough that the effect is measurable with good precision, but it must not be made so large that higher order concentration terms are required in Eq. (1). Since SEC columns dilute the sample by about an order of magnitude per column, it is usually the case that the concentrations resulting from chromatographic separations are small enough that A2 can usually be neglected. Details of the chromatographic separation methods, the definitions and calculations of the mass and size moments, and an explanation of the terminology used to describe the associated distributions may be found in the 1993 review article by Wyatt in Analytica Chimica Acta, volume 272, pages 1 through 40.
In summary, there are two modes of light scattering measurements. In the batch mode, a series of light scattering measurements of a single sample are made at different concentrations. The concentration and angular dependence of the scattering signals allows Mw,       ⟨          r      g      2        ⟩    ,
and A2 to be determined. In the chromatography mode, the sample composition changes as the sample elutes, so a priori knowledge of A2 is required, but the distributions M(v) and rg2 (v) can be measured. From the distributions, the averages Mw, and   ⟨      r    g    2    ⟩
may be calculated. It should be noted that the values calculated from the fractionated sample measurements should be identical to the values measured from batch samples. Discrepancies arise from the assumption of A2=0 and the distortions due to interdetector band broadening.
Although A2 can be determined from batch measurements, the question remains; can the second virial coefficient be measured accurately when the sample concentration is changing continuously in time, as is the case for a chromatographic elution? In U.S. Pat. No. 5,129,723 by Howie, Jackson, and Wyatt, a method was described whereby an unfractionated sample was injected into a MALS detector following dilution and thorough mixing. This procedure produced a sample peak passing through the light scattering detector whose profile was assumed to be proportional to the concentration profile of the diluted, yet unfractionated, sample. Since the mass distribution at each slice was the same, it was assumed that each point of the profile was proportional, at that point, to the sample""s concentration times the weight averaged molar mass by referring to Eq. (3) and setting A2=0. On this basis, a Zimm plot could be produced using a set of these points, and the associated weight average molar mass, mean square radius, and second virial coefficient were then derived. It was thought that a concentration detector was not needed, since knowledge of the total mass injected was sufficient to convert the sample peak curve into a concentration profile. However, the method was flawed because the assumption that A2 was zero contradicted the derived result that it was not.
A second method is to use a chromatography configuration, in which both a light scattering and a concentration detector are used. If one injects a monodisperse sample, or develops the chromatography method to fractionate peaks that are monomeric, the weight average molar mass at each eluting fraction should be constant throughout each peak. From the MALS and concentration data, a Zimm plot analysis may be performed from values at several different slices or sets of slices of the elution profile. The weight average molar mass, mean square radius, and second virial coefficient then may be derived. However, for most proteins, the mean square radius is too small to be accurately measured.
While this method may work in principle, there are practical difficulties that prevent it from being generally applicable. The experimental setup described above requires two detectors. Since the fluid must pass through capillaries and unions as it travels between the detectors, mixing and diffusion give rise to xe2x80x9cinterdetector band broadeningxe2x80x9d. The downstream peak is xe2x80x9cbroaderxe2x80x9d than the upstream peak. This means that a monodisperse sample will produce a measured mass distribution that appears polydisperse, unless the broadening effect is taken into account. This effect is particularly problematic for proteins, since their FOMs are typically much less than one and the second virial coefficient produces a small contribution to scattered light. The error associated with band broadening usually dominates, and the derived second virial coefficient is significantly distorted. Various analytical corrections of band broadening have been developed over the years, but since the A2 contribution to the light scattering signals is so small, the resulting values of A2 depend sensitively on the exact model of band broadening used. This sensitive dependence on model has made this method unreliable.
Since the protein molar mass for a monodisperse sample in a suitable buffering solvent is often easily measured by MALS, mass spectroscopy, or direct sequencing, the U.S. Pat. No. 6,411,383, by Wyatt entitled xe2x80x9cMethod for measuring the second virial coefficient.xe2x80x9d emphasized the utility of circumventing the creation of the multiple concentrations required to make a full Zimm plot. However, it will be shown that when the methods described therein are extended to the more general case of multiple concentration aliquots, the method no longer requires a priori knowledge of the molar mass. Additionally the accuracy is improved due to the averaging achieved by the extended method.
Many scientists utilize on-line methods for making Zimm plots, and the afore-referenced ASTRA(copyright) program makes the determination far simpler than earlier manual methods. Currently, the best means for measuring A2 of an unfractionated sample is to prepare aliquots at different concentrations and to inject them sequentially, using a syringe or syringe pump, directly into the light scattering cell of a MALS instrument. Relative to the baseline scattering of the pure solvent, each aliquot produces a ramp up to a plateau as the injection fills the cell. From a selected range of points, collectively referred to as a peak, on each of the plateaus, the software accepts the corresponding entries of the prepared concentrations and generates a Zimm plot. This approach has several shortcomings, not the least of which is the need to prepare and use relatively large amounts of sample. When injecting into a flow cell, the need to produce flat plateaus means that the cell must be overfilled several times. For a flow cell with an internal volume of 80 xcexcl, upward of 500 xcexcl of sample are required for each aliquot. For modem pharmaceutical research, this requirement makes the current technique infeasible. Often, this quantity of sample cannot practically be synthesized. In addition, for proteins and similar biopolymers, each injected sample must be dialyzed before injection. This adds significantly to the preparation time and labor required to make the measurement.
Lastly, in this invention, we show that by integrating across the peaks, the dependence on the peak shape is eliminated, and the band broadening correction reduces to a single parameter. The sensitive dependence on the model of broadening is removed making the method practical and reliable. Not only may A2 be measured with great ease in the presence of band broadening for monodisperse samples such as proteins, but also for the case of unfractionated samples injected directly into a light scattering measurement cell.
It is a major objective of this invention to extend the methods described in the parent invention to provide a method to extract Mw,       ⟨          r      g      2        ⟩    ,
and A2 directly from a sequence of injections with different concentrations, thus eliminating the need to know Mw, a priori. We also disclose two fitting techniques that extract these parameters. One method is based on a global nonlinear least squares fit the data. The second analysis technique is modeled after the graphical method of Zimm. The resulting plots will be called Trainoff-Wyatt, TW, plots to differentiate them explicitly from Zimm plots. It is an objective of this invention to automate the process and eliminate, as much as possible, the manual sample preparation. Another objective of this invention is to perform the measurement more accurately by exploiting the averaging implicit in the multi-injection technique. Lastly it is an objective of this invention to make such determinations using minimal quantities of samples.
The aforementioned U.S. Pat. No. 6,411,383 by Wyatt concerns a method by which the second virial coefficient of a sample comprised of a monodisperse molar mass distribution in a solvent could be determined directly from a single injection. The same method could be applied also to certain classes of unfractionated samples. The present invention is directed to the more general application of using a series of concentration injections. The Wyatt patent requires an a priori knowledge of the sample""s weight average molar mass. This invention determines Mw,       ⟨          r      g      2        ⟩    ,
and A2 using an analysis technique analogous to that presented by Zimm.
Additionally, the Wyatt patent assumes that the interdetector band broadening is negligible. It is often desirable to minimize the quantity of sample used in the measurement. Therefore small injections are used, giving rise to narrow peaks. In this case, interdetector band broadening cannot be neglected. In this invention, we present several methods to account for this effect.
The method begins with the preparation of a set of j dilutions of the sample for injection onto a SEC column set. If an auto sampler is available, then a single sample may be provided for subsequent automatic dilution to specified concentrations. There are two ways that that an autosampler may be used to prepare the dilutions. First it can inject progressively smaller quantities of the sample into the injection loop, under-filling it. Alternatively, the autosampler can pre-dilute the sample before filling the injection loop, and then fill it completely. The latter method is preferred since it will give rise to congruent concentration profiles, with different amplitudes, for each injection.
The analysis requires a set of decreasing concentrations of the sample to be measured. For the case of proteins, each sample aliquot must be dialyzed. Introducing a set of SEC columns through which each sample will flow obviates this requirement, as dialysis will occur during the passage of the sample through the column. Following the columns, a MALS detector and concentration detector are connected serially. For small molecules whose mean square radii are too small to measure, measurement at a single scattering angle such as 90xc2x0 may suffice, though the precision of the determination would be diminished. These are the conventional elements of a standard separation by SEC resulting in an absolute determination of the eluting molar masses present in the sample. If neither dialysis nor fractionation is required, the separation columns can be eliminated, and the sample may be injected directly into the flow cell of the light scattering instrument. Alternatively, a guard column may be used to achieve dialysis without separation.